Since an arbitrary 4-manifold with a metric does not naturally come equipped with a (possibly complex-valued)
almost para-Hermitian structure, we first focus on classifying all possible such structures.
We show that regardless of signature, an almost para-Hermitian structure
is essentially equivalent to an (anti-)self-dual 2-form (in the sense of Hodge duality).
Then we use this result to describe the space \(P_{x}\)  of para-Hermitian structures at any point \(x\in M\) :
we show that \(P_{x}\)  has two connected components and, imposing appropriate reality conditions whenever possible,
each component can be described as:
a complex projective line \(\mathbb {CP}^{1}\)  in Riemannian signatureThis turns out to be analogous
to the result on almost-complex structures, which is already known in the literature [1]}.,
the space \((\mathbb {RP}^{1}\times \mathbb {RP}^{1})\backslash \mathbb {RP}^{1}\)  in split signature,
and the space \((\mathbb {CP}^{1}\times \mathbb {CP}^{1})\backslash \mathbb {CP}^{1}\)  in Lorentz signature
(and in complex 4-manifolds).
We show that, topologically, these spaces are respectively
a real 2-sphere \(S^{2}\) , a hyperboloid of one sheet \(H^{1}\) , and a complex 2-sphere \(\mathbb {C}S^{2}\) .
For Hermitian structures, the corresponding spaces are \(\mathbb {CP}^{1}\) ,
\(\mathbb {CP}^{1}\backslash \mathbb {RP}^{1}\)  (a hyperboloid of two sheets)
and \((\mathbb {CP}^{1}\times \mathbb {CP}^{1})\backslash \mathbb {CP}^{1}\) , respectively.
We also find that there is a natural “gauge freedom” associated to the representation of a (para-)Hermitian structure,
and we develop a correspondingly covariant formalism.
